1 Introduction

General Relativity (GR) [225226Jump To The Next Citation Point] is widely accepted as a fundamental theory to describe the geometric properties of spacetime. In a homogeneous and isotropic spacetime the Einstein field equations give rise to the Friedmann equations that describe the evolution of the universe. In fact, the standard big-bang cosmology based on radiation and matter dominated epochs can be well described within the framework of General Relativity.

However, the rapid development of observational cosmology which started from 1990s shows that the universe has undergone two phases of cosmic acceleration. The first one is called inflation [564Jump To The Next Citation Point339Jump To The Next Citation Point291Jump To The Next Citation Point524Jump To The Next Citation Point], which is believed to have occurred prior to the radiation domination (see [402391Jump To The Next Citation Point71Jump To The Next Citation Point] for reviews). This phase is required not only to solve the flatness and horizon problems plagued in big-bang cosmology, but also to explain a nearly flat spectrum of temperature anisotropies observed in Cosmic Microwave Background (CMB) [541]. The second accelerating phase has started after the matter domination. The unknown component giving rise to this late-time cosmic acceleration is called dark energy [310] (see [517141480485171Jump To The Next Citation Point32] for reviews). The existence of dark energy has been confirmed by a number of observations – such as supernovae Ia (SN Ia) [490506507], large-scale structure (LSS) [577Jump To The Next Citation Point578Jump To The Next Citation Point], baryon acoustic oscillations (BAO) [227Jump To The Next Citation Point487], and CMB [560561Jump To The Next Citation Point367Jump To The Next Citation Point].

These two phases of cosmic acceleration cannot be explained by the presence of standard matter whose equation of state w = P ∕ρ satisfies the condition w ≥ 0 (here P and ρ are the pressure and the energy density of matter, respectively). In fact, we further require some component of negative pressure, with w < − 1∕3, to realize the acceleration of the universe. The cosmological constant Λ is the simplest candidate of dark energy, which corresponds to w = − 1. However, if the cosmological constant originates from a vacuum energy of particle physics, its energy scale is too large to be compatible with the dark energy density [614]. Hence we need to find some mechanism to obtain a small value of Λ consistent with observations. Since the accelerated expansion in the very early universe needs to end to connect to the radiation-dominated universe, the pure cosmological constant is not responsible for inflation. A scalar field ϕ with a slowly varying potential can be a candidate for inflation as well as for dark energy.

Although many scalar-field potentials for inflation have been constructed in the framework of string theory and supergravity, the CMB observations still do not show particular evidence to favor one of such models. This situation is also similar in the context of dark energy – there is a degeneracy as for the potential of the scalar field (“quintessence” [111634267263615503257155]) due to the observational degeneracy to the dark energy equation of state around w = − 1. Moreover it is generally difficult to construct viable quintessence potentials motivated from particle physics because the field mass responsible for cosmic acceleration today is very small (− 33 m ϕ ≃ 10 eV[140365].

While scalar-field models of inflation and dark energy correspond to a modification of the energy-momentum tensor in Einstein equations, there is another approach to explain the acceleration of the universe. This corresponds to the modified gravity in which the gravitational theory is modified compared to GR. The Lagrangian density for GR is given by f(R ) = R − 2Λ, where R is the Ricci scalar and Λ is the cosmological constant (corresponding to the equation of state w = − 1). The presence of Λ gives rise to an exponential expansion of the universe, but we cannot use it for inflation because the inflationary period needs to connect to the radiation era. It is possible to use the cosmological constant for dark energy since the acceleration today does not need to end. However, if the cosmological constant originates from a vacuum energy of particle physics, its energy density would be enormously larger than the today’s dark energy density. While the Λ-Cold Dark Matter (ΛCDM) model (f(R ) = R − 2Λ) fits a number of observational data well [367Jump To The Next Citation Point368], there is also a possibility for the time-varying equation of state of dark energy [1011450451Jump To The Next Citation Point630].

One of the simplest modifications to GR is the f (R) gravity in which the Lagrangian density f is an arbitrary function of R [77512102106]. There are two formalisms in deriving field equations from the action in f (R) gravity. The first is the standard metric formalism in which the field equations are derived by the variation of the action with respect to the metric tensor gμν. In this formalism the affine connection α Γβγ depends on gμν. Note that we will consider here and in the remaining sections only torsion-free theories. The second is the Palatini formalism [481Jump To The Next Citation Point] in which gμν and Γ αβγ are treated as independent variables when we vary the action. These two approaches give rise to different field equations for a non-linear Lagrangian density in R, while for the GR action they are identical with each other. In this article we mainly review the former approach unless otherwise stated. In Section 9 we discuss the Palatini formalism in detail.

The model with f(R ) = R + αR2 (α > 0) can lead to the accelerated expansion of the Universe because of the presence of the αR2 term. In fact, this is the first model of inflation proposed by Starobinsky in 1980 [564Jump To The Next Citation Point]. As we will see in Section 7, this model is well consistent with the temperature anisotropies observed in CMB and thus it can be a viable alternative to the scalar-field models of inflation. Reheating after inflation proceeds by a gravitational particle production during the oscillating phase of the Ricci scalar [565Jump To The Next Citation Point606Jump To The Next Citation Point426Jump To The Next Citation Point].

The discovery of dark energy in 1998 also stimulated the idea that cosmic acceleration today may originate from some modification of gravity to GR. Dark energy models based on f (R) theories have been extensively studied as the simplest modified gravity scenario to realize the late-time acceleration. The model with a Lagrangian density f (R ) = R − α ∕Rn (α > 0,n > 0) was proposed for dark energy in the metric formalism [113Jump To The Next Citation Point120Jump To The Next Citation Point114Jump To The Next Citation Point143Jump To The Next Citation Point456Jump To The Next Citation Point]. However it was shown that this model is plagued by a matter instability [215Jump To The Next Citation Point244Jump To The Next Citation Point] as well as by a difficulty to satisfy local gravity constraints [469Jump To The Next Citation Point470Jump To The Next Citation Point245Jump To The Next Citation Point233Jump To The Next Citation Point154Jump To The Next Citation Point448Jump To The Next Citation Point134Jump To The Next Citation Point]. Moreover it does not possess a standard matter-dominated epoch because of a large coupling between dark energy and dark matter [28Jump To The Next Citation Point29Jump To The Next Citation Point]. These results show how non-trivial it is to obtain a viable f (R) model. Amendola et al. [26Jump To The Next Citation Point] derived conditions for the cosmological viability of f (R) dark energy models. In local regions whose densities are much larger than the homogeneous cosmological density, the models need to be close to GR for consistency with local gravity constraints. A number of viable f (R) models that can satisfy both cosmological and local gravity constraints have been proposed in . [26Jump To The Next Citation Point382Jump To The Next Citation Point31Jump To The Next Citation Point306Jump To The Next Citation Point568Jump To The Next Citation Point35Jump To The Next Citation Point587Jump To The Next Citation Point206164396Jump To The Next Citation Point]. Since the law of gravity gets modified on large distances in f (R) models, this leaves several interesting observational signatures such as the modification to the spectra of galaxy clustering [146Jump To The Next Citation Point74Jump To The Next Citation Point544Jump To The Next Citation Point526Jump To The Next Citation Point251Jump To The Next Citation Point597Jump To The Next Citation Point493Jump To The Next Citation Point], CMB [627544Jump To The Next Citation Point382Jump To The Next Citation Point545Jump To The Next Citation Point], and weak lensing [595Jump To The Next Citation Point528Jump To The Next Citation Point]. In this review we will discuss these topics in detail, paying particular attention to the construction of viable f (R) models and resulting observational consequences.

The f (R) gravity in the metric formalism corresponds to generalized Brans–Dicke (BD) theory [100Jump To The Next Citation Point] with a BD parameter ωBD = 0 [467Jump To The Next Citation Point579Jump To The Next Citation Point152Jump To The Next Citation Point]. Unlike original BD theory [100Jump To The Next Citation Point], there exists a potential for a scalar-field degree of freedom (called “scalaron” [564Jump To The Next Citation Point]) with a gravitational origin. If the mass of the scalaron always remains as light as the present Hubble parameter H0, it is not possible to satisfy local gravity constraints due to the appearance of a long-range fifth force with a coupling of the order of unity. One can design the field potential of f (R) gravity such that the mass of the field is heavy in the region of high density. The viable f (R) models mentioned above have been constructed to satisfy such a condition. Then the interaction range of the fifth force becomes short in the region of high density, which allows the possibility that the models are compatible with local gravity tests. More precisely the existence of a matter coupling, in the Einstein frame, gives rise to an extremum of the effective field potential around which the field can be stabilized. As long as a spherically symmetric body has a “thin-shell” around its surface, the field is nearly frozen in most regions inside the body. Then the effective coupling between the field and non-relativistic matter outside the body can be strongly suppressed through the chameleon mechanism [344Jump To The Next Citation Point343Jump To The Next Citation Point]. The experiments for the violation of equivalence principle as well as a number of solar system experiments place tight constraints on dark energy models based on f (R) theories [306Jump To The Next Citation Point251Jump To The Next Citation Point587Jump To The Next Citation Point134Jump To The Next Citation Point101Jump To The Next Citation Point].

The spherically symmetric solutions mentioned above have been derived under the weak gravity backgrounds where the background metric is described by a Minkowski space-time. In strong gravitational backgrounds such as neutron stars and white dwarfs, we need to take into account the backreaction of gravitational potentials to the field equation. The structure of relativistic stars in f (R) gravity has been studied by a number of authors [349Jump To The Next Citation Point350Jump To The Next Citation Point594Jump To The Next Citation Point43Jump To The Next Citation Point600Jump To The Next Citation Point46642Jump To The Next Citation Point167Jump To The Next Citation Point]. Originally the difficulty of obtaining relativistic stars was pointed out in [349Jump To The Next Citation Point] in connection to the singularity problem of f (R) dark energy models in the high-curvature regime [266Jump To The Next Citation Point]. For constant density stars, however, a thin-shell field profile has been analytically derived in [594Jump To The Next Citation Point] for chameleon models in the Einstein frame. The existence of relativistic stars in f (R) gravity has been also confirmed numerically for the stars with constant [43Jump To The Next Citation Point600Jump To The Next Citation Point] and varying [42Jump To The Next Citation Point] densities. In this review we shall also discuss this issue.

It is possible to extend f (R) gravity to generalized BD theory with a field potential and an arbitrary BD parameter ωBD. If we make a conformal transformation to the Einstein frame [213Jump To The Next Citation Point609Jump To The Next Citation Point408Jump To The Next Citation Point611Jump To The Next Citation Point249Jump To The Next Citation Point268Jump To The Next Citation Point], we can show that BD theory with a field potential corresponds to the coupled quintessence scenario [23Jump To The Next Citation Point] with a coupling Q between the field and non-relativistic matter. This coupling is related to the BD parameter via the relation 1∕(2Q2 ) = 3 + 2 ωBD [343Jump To The Next Citation Point596Jump To The Next Citation Point]. One can recover GR by taking the limit Q → 0, i.e., ωBD → ∞. The f (R) gravity in the metric formalism corresponds to -- Q = − 1∕√ 6 [28Jump To The Next Citation Point], i.e., ωBD = 0. For large coupling models with |Q | = 𝒪 (1) it is possible to design scalar-field potentials such that the chameleon mechanism works to reduce the effective matter coupling, while at the same time the field is sufficiently light to be responsible for the late-time cosmic acceleration. This generalized BD theory also leaves a number of interesting observational and experimental signatures [596Jump To The Next Citation Point].

In addition to the Ricci scalar R, one can construct other scalar quantities such as R μνR μν and R Rμνρσ μνρσ from the Ricci tensor R μν and Riemann tensor R μνρσ [142Jump To The Next Citation Point]. For the Gauss–Bonnet (GB) curvature invariant defined by 2 αβ αβγδ 𝒢 ≡ R − 4R αβR + R αβγδR, it is known that one can avoid the appearance of spurious spin-2 ghosts [572Jump To The Next Citation Point67Jump To The Next Citation Point302Jump To The Next Citation Point] (see also [98465Jump To The Next Citation Point153Jump To The Next Citation Point447Jump To The Next Citation Point110Jump To The Next Citation Point181Jump To The Next Citation Point109Jump To The Next Citation Point]). In order to give rise to some contribution of the GB term to the Friedmann equation, we require that (i) the GB term couples to a scalar field ϕ, i.e., F (ϕ)𝒢 or (ii) the Lagrangian density f is a function of 𝒢, i.e., f(𝒢 ). The GB coupling in the case (i) appears in low-energy string effective action [275Jump To The Next Citation Point] and cosmological solutions in such a theory have been studied extensively (see [34Jump To The Next Citation Point273Jump To The Next Citation Point105Jump To The Next Citation Point147Jump To The Next Citation Point588Jump To The Next Citation Point409468] for the construction of nonsingular cosmological solutions and [463Jump To The Next Citation Point360Jump To The Next Citation Point361Jump To The Next Citation Point593Jump To The Next Citation Point523Jump To The Next Citation Point452Jump To The Next Citation Point453Jump To The Next Citation Point381Jump To The Next Citation Point25Jump To The Next Citation Point] for the application to dark energy). In the case (ii) it is possible to construct viable models that are consistent with both the background cosmological evolution and local gravity constraints [458Jump To The Next Citation Point188Jump To The Next Citation Point189Jump To The Next Citation Point] (see also [165Jump To The Next Citation Point180Jump To The Next Citation Point178Jump To The Next Citation Point383Jump To The Next Citation Point633Jump To The Next Citation Point599]). However density perturbations in perfect fluids exhibit negative instabilities during both the radiation and the matter domination, irrespective of the form of f(𝒢) [383Jump To The Next Citation Point182Jump To The Next Citation Point]. This growth of perturbations gets stronger on smaller scales, which is difficult to be compatible with the observed galaxy spectrum unless the deviation from GR is very small. We shall review such theories as well as other modified gravity theories.

This review is organized as follows. In Section 2 we present the field equations of f (R) gravity in the metric formalism. In Section 3 we apply f (R) theories to the inflationary universe. Section 4 is devoted to the construction of cosmologically viable f (R) dark energy models. In Section 5 local gravity constraints on viable f (R) dark energy models will be discussed. In Section 6 we provide the equations of linear cosmological perturbations for general modified gravity theories including metric f (R) gravity as a special case. In Section 7 we study the spectra of scalar and tensor metric perturbations generated during inflation based on f (R) theories. In Section 8 we discuss the evolution of matter density perturbations in f (R) dark energy models and place constraints on model parameters from the observations of large-scale structure and CMB. Section 9 is devoted to the viability of the Palatini variational approach in f (R) gravity. In Section 10 we construct viable dark energy models based on BD theory with a potential as an extension of f (R) theories. In Section 11 the structure of relativistic stars in f (R) theories will be discussed in detail. In Section 12 we provide a brief review of Gauss–Bonnet gravity and resulting observational and experimental consequences. In Section 13 we discuss a number of other aspects of f (R) gravity and modified gravity. Section 14 is devoted to conclusions.

There are other review articles on f (R) gravity [556Jump To The Next Citation Point555618] and modified gravity [171Jump To The Next Citation Point459126397217]. Compared to those articles, we put more weights on observational and experimental aspects of f (R) theories. This is particularly useful to place constraints on inflation and dark energy models based on f (R) theories. The readers who are interested in the more detailed history of f (R) theories and fourth-order gravity may have a look at the review articles by Schmidt [531] and Sotiriou and Faraoni [556Jump To The Next Citation Point].

In this review we use units such that c = ℏ = kB = 1, where c is the speed of light, ℏ is reduced Planck’s constant, and k B is Boltzmann’s constant. We define κ2 = 8πG = 8π∕m2 = 1∕M 2 pl pl, where G is the gravitational constant, 19 mpl = 1.22 × 10 GeV is the Planck mass with a reduced value √ --- Mpl = mpl∕ 8π = 2.44 × 1018 GeV. Throughout this review, we use a dot for the derivative with respect to cosmic time t and “,X” for the partial derivative with respect to the variable X, e.g., f,R ≡ ∂f ∕∂R and f,RR ≡ ∂2f ∕∂R2. We use the metric signature (− ,+, +, + ). The Greek indices μ and ν run from 0 to 3, whereas the Latin indices i and j run from 1 to 3 (spatial components).

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